(264e) Cfd-Based Shape Optimization of Pressure-Driven Microchannels Via Adjoint Formulation
AIChE Annual Meeting
Tuesday, November 18, 2008 - 1:50pm to 2:10pm
With the advances in computational resources and algorithms, optimal shape design based on computational fluid dynamics (CFD) is an interesting filed for industrial applications. Although gradient based method is widely used to solve optimization problems, it requires the solution of a large linear system of equations to calculate the sensitivity of the cost function with respect to each design parameter. Thus, for n design parameters, n large linear programming problems have to be solved. Consequently, gradient based method requires enormous computation time when the number of design variables is large. Recently, the adjoint variable method attracts the attention as an efficient sensitivity analysis method, particularly for aeronautical design, since it allows to successfully obtain the shape gradient functions independently of the number of design variables.
In this research, an automatic shape optimization system based on the adjoint variable method is developed using C language on Windows platform. The flow equation is added to the cost function by introducing the set of Lagrange multipliers, because it must be fulfilled by the optimal solution. The principal steps of the optimization procedure are as follows: 1) assume an initial shape, 2) generate computational grids, 3) solve the flow equations, viz. the Navier-Stokes equation and the continuity equation, for deriving the flow velocity and the pressure, 4) solve the adjoint equation to obtain the set of Lagrange multipliers, 5) calculate the shape gradient functions, 6) obtain a new shape by moving each point on the boundary, and 7) go to step 2 unless the change in the cost function is smaller than a desired convergence parameter. Each design cycle requires the numerical solution of both the flow and the adjoint equations, whose computational time is roughly twice of that required to obtain the flow solution. In order to validate the effectiveness of the developed system, the optimal shape design problems of the pressure-driven microchannels are solved. The shape of the microchannels is an important design variable to achieve the desired function. As examples, pressure drop minimization problems of a 180° curved channel and a branched channel in incompressible flows under constant volume conditions are formulated. The atmospheric pressure is set at the outflow boundary, and a uniform velocity profile (Re = 10) is prescribed at the inflow boundary. 92 rounds of design cycles are required till convergence for 42 design variables, which are assigned on the curved channel surface. Each design cycle requires approximately 10 seconds. The pressure drop of the optimally designed curved channel is decreased by 27.6 % as compared with that of the initial shape. It is demonstrated that the adjoint variable method can be used to formulate computationally feasible procedures for the shape design of microchannels. Our future work will focus on the extension of the developed system to shape optimization problems of thermo-fluidic microdevices.